Formula

Curved and total surface of a cylinder

Mensuration • Solids • Cylinder

Formula

CSA = 2 π r h, TSA = 2 π r (h + r)

A right circular cylinder of radius

r

and height

h

has curved surface area

2 π r h

— the rectangle it unrolls into — and total surface area

2 π r h + 2 π r^{2} = 2 π r (h + r)

when both circular ends are closed.

How to use it

1Decide first how many ends are present: a pipe has none, a cup or a tank open at the top has one, a closed tin has two.
2The ratio $CSA : TSA = h : (h + r)$ , so a stated ratio of the two surfaces is really a condition on the shape.
3The curved surface is the base circumference times the height, which is why unrolling it gives a plain rectangle.

Worked example

A closed cylindrical tin has radius

7

cm and height

10

cm. Find its total surface area.

Step-by-step

1
$TSA = 2 π r (h + r) = 2 \times \frac{22}{7} \times 7 \times (10 + 7)$
Why: Both ends are closed, so both circles count.
2
$= 44 \times 17 = 748$ cm $^{2}$

Answer

748

^{2}

Watch out

Adding only one end, or none, is the commonest error, and a well-built option list prints those partial totals. The curved surface never includes the ends, whatever the vessel is called.

Builds on

Area and circumference of a circle

Also known as

curved surface area of a cylinder, total surface area of a cylinder, 2 pi r h, lateral surface of a cylinder

Appears in past papers

Formula

Worked example

Appears in past papers

2020 (I)

2019 (II)

2019 (I)

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2018 (I)

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2017 (I)

2016 (II)